{
  "id": "2302.01865",
  "version": "v1",
  "published": "2023-02-03T17:09:09.000Z",
  "updated": "2023-02-03T17:09:09.000Z",
  "title": "On a Weighted Series of the Hurwitz Zeta Function",
  "authors": [
    "Matthew Fox",
    "Chaitanya Karamchedu"
  ],
  "comment": "8 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this note we prove that for all $a \\in \\mathbb{N}$, $x \\in \\mathbb{R}_+ \\cup \\{0\\}$, and $s \\in \\mathbb{C}$ with $\\Re(s) > a + 2$, the (alternating) weighted series of the Hurwitz zeta function, $$ \\sum_{k \\geq 1} (\\pm 1)^k (k + x)^a\\zeta(s,k + x), $$ resolves into a finite combination of Hurwitz (Lerch) zeta functions. This applies in Marichal and Zena\\\"idi's theory on analogues of the Bohr-Mollerup theorem for higher-order convex functions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-03T17:09:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11M35"
    ],
    "keywords": [
      "hurwitz zeta function",
      "weighted series",
      "higher-order convex functions",
      "bohr-mollerup theorem",
      "finite combination"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}